论文标题
McKean-Vlasov SDES的熵和Wasserstein距离的指数收敛
Exponential Convergence in Entropy and Wasserstein Distance for McKean-Vlasov SDEs
论文作者
论文摘要
证明了以下类型的指数融合，用于（非分级或退化）McKean-Vlasov Sdes：$$ W_2（μ__T，μ__\ infty）^2 +{\ rm ent}（μ_t|μ_t|μ__\ \ \ \ iffty） \min\big\{W_2(μ_0, μ_\infty)^2,{\rm Ent}(μ_0|μ_\infty)\big\},\ \ t\ge 1,$$ where $c,λ>0$ are constants, $μ_t$ is the distribution of the solution at time $t$, $μ_\infty$ is the unique不变的概率度量，$ {\ rm ent} $是相对熵，$ w_2 $是$ l^2 $ -wasserstein距离。特别是，这种类型的指数收敛适用于某些（非定位或退化）颗粒介质类型方程，从而在平均场熵中的指数收敛上概括了在[CMV，GLW]中所研究的介质方程。
The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(μ_t,μ_\infty)^2 +{\rm Ent}(μ_t|μ_\infty)\le c {\rm e}^{-λt} \min\big\{W_2(μ_0, μ_\infty)^2,{\rm Ent}(μ_0|μ_\infty)\big\},\ \ t\ge 1,$$ where $c,λ>0$ are constants, $μ_t$ is the distribution of the solution at time $t$, $μ_\infty$ is the unique invariant probability measure, ${\rm Ent}$ is the relative entropy and $W_2$ is the $L^2$-Wasserstein distance. In particular, this type exponential convergence holds for some (non-degenerate or degenerate) granular media type equations generalizing those studied in [CMV, GLW] on the exponential convergence in a mean field entropy.
